(b) On what intervals of x in the form does the series converge uniformly?

Now since \{ }, by M-Weierstrass Test the series converges uniformly on and .

Be careful, you need a uniform bound for for applying M-test. Your argument is correct for the compact subsets of , when such a bound always exists.

Just observe that this gives you an answer for (d), for each you have that there exists such that the series converges uniformly on .
Now, recall that the uniform convergence of continuous functions preserves continuity. Actually M-test gives continuity whenever the terms of the series are.

Be careful, you need a uniform bound for for applying M-test. Your argument is correct for the compact subsets of , when such a bound always exists.

Just observe that this gives you an answer for (d), for each you have that there exists such that the series converges uniformly on .
Now, recall that the uniform convergence of continuous functions preserves continuity. Actually M-test gives continuity whenever the terms of the series are.

What do you mean by uniform bound?
Also, I can't quite seem to get my head around (d), why is it that uniform convergence guaratees preservation of continuity?

An uniform bound on [a,b] (or (a,b) is a bound for every . It is a necessary condition in M test that for each and being convergent. Now look that in an interval (0,b) you have
that the bound depends on and goes to infinity as goes to 0. This is not allowed in the M-test, you need a bound independent of , i.e. valid for all the interval.

The fact of that uniform convergence of a sequence of continuous functions is again continuous is a well known theorem, I don't remember the name of the author. That is because C(K) endowed with the uniform norm is a Banach space if K is compact. Sure that you can check it in a book of Analysis, I suppose in Apostol for example.